This makes for a very interesting discussion.
My own suspicion was that the Four Quadrant model of the galaxy was developed between TOS and STVI, possibly as a 'short hand' when more of the immediate region of the galaxy become familiar.
From STVI the premise seems that Beta Quadrant isn't too far from the Federation and/or the Klingon Empire.
But I also got the impression the Beta Quadrant was less-explored than the Alpha (which by the very nature of things must be overwhelmingly unexplored). I'm puzzled about any canon sources that explicitly say any polity we've heard of lies within the Beta Quadrant. Are there any?
Why are the lines where they are? No way to tell. Could easily be the detection of two identical weird events/objects on either side of the galactic center, or something a lot more esoteric/vague. Could even be a matter of dispute about exactly where the lines go, amidst general agreement over the bulk of what constitutes each (not unlike the "Bermuda Triangle" or "the Middle East" or even "The Bible Belt").
I've also presume a fair number of items we've heard of that seem from their description just too far way--the "edge of the galaxy" or "the center of the galaxy"--could conceivably be poor renditions of names given features. After all, "Greenland" isn't really. The English Channel might as well be called the French, while the Near East has precious little to do with geography per se. The New World certainly isn't a new planet. And the Old World is exactly the same age as the Americas.